# Properties

 Label 7056.2.k.b Level $7056$ Weight $2$ Character orbit 7056.k Analytic conductor $56.342$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7056 = 2^{4} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7056.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$56.3424436662$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{5} +O(q^{10})$$ $$q + \beta_{3} q^{5} -\beta_{1} q^{11} -3 \beta_{2} q^{13} + 2 \beta_{3} q^{17} -\beta_{2} q^{19} -4 \beta_{1} q^{23} + q^{25} -2 \beta_{1} q^{29} + \beta_{2} q^{31} - q^{37} -3 \beta_{3} q^{41} + q^{43} + 5 \beta_{3} q^{47} -2 \beta_{1} q^{53} -2 \beta_{2} q^{55} -2 \beta_{3} q^{59} -2 \beta_{2} q^{61} -9 \beta_{1} q^{65} -11 q^{67} + 5 \beta_{1} q^{71} -\beta_{2} q^{73} -5 q^{79} -3 \beta_{3} q^{83} + 12 q^{85} -2 \beta_{3} q^{89} -3 \beta_{1} q^{95} -6 \beta_{2} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 4q^{25} - 4q^{37} + 4q^{43} - 44q^{67} - 20q^{79} + 48q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 1$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 4 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 1$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/7056\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1765$$ $$4609$$ $$6175$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
881.1
 −1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i 1.22474 − 0.707107i
0 0 0 −2.44949 0 0 0 0 0
881.2 0 0 0 −2.44949 0 0 0 0 0
881.3 0 0 0 2.44949 0 0 0 0 0
881.4 0 0 0 2.44949 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7056.2.k.b 4
3.b odd 2 1 inner 7056.2.k.b 4
4.b odd 2 1 441.2.c.a 4
7.b odd 2 1 inner 7056.2.k.b 4
7.c even 3 1 1008.2.bt.b 4
7.d odd 6 1 1008.2.bt.b 4
12.b even 2 1 441.2.c.a 4
21.c even 2 1 inner 7056.2.k.b 4
21.g even 6 1 1008.2.bt.b 4
21.h odd 6 1 1008.2.bt.b 4
28.d even 2 1 441.2.c.a 4
28.f even 6 1 63.2.p.a 4
28.f even 6 1 441.2.p.a 4
28.g odd 6 1 63.2.p.a 4
28.g odd 6 1 441.2.p.a 4
84.h odd 2 1 441.2.c.a 4
84.j odd 6 1 63.2.p.a 4
84.j odd 6 1 441.2.p.a 4
84.n even 6 1 63.2.p.a 4
84.n even 6 1 441.2.p.a 4
140.p odd 6 1 1575.2.bk.c 4
140.s even 6 1 1575.2.bk.c 4
140.w even 12 2 1575.2.bc.a 8
140.x odd 12 2 1575.2.bc.a 8
252.n even 6 1 567.2.s.d 4
252.o even 6 1 567.2.s.d 4
252.r odd 6 1 567.2.i.d 4
252.u odd 6 1 567.2.i.d 4
252.bb even 6 1 567.2.i.d 4
252.bj even 6 1 567.2.i.d 4
252.bl odd 6 1 567.2.s.d 4
252.bn odd 6 1 567.2.s.d 4
420.ba even 6 1 1575.2.bk.c 4
420.be odd 6 1 1575.2.bk.c 4
420.bp odd 12 2 1575.2.bc.a 8
420.br even 12 2 1575.2.bc.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.p.a 4 28.f even 6 1
63.2.p.a 4 28.g odd 6 1
63.2.p.a 4 84.j odd 6 1
63.2.p.a 4 84.n even 6 1
441.2.c.a 4 4.b odd 2 1
441.2.c.a 4 12.b even 2 1
441.2.c.a 4 28.d even 2 1
441.2.c.a 4 84.h odd 2 1
441.2.p.a 4 28.f even 6 1
441.2.p.a 4 28.g odd 6 1
441.2.p.a 4 84.j odd 6 1
441.2.p.a 4 84.n even 6 1
567.2.i.d 4 252.r odd 6 1
567.2.i.d 4 252.u odd 6 1
567.2.i.d 4 252.bb even 6 1
567.2.i.d 4 252.bj even 6 1
567.2.s.d 4 252.n even 6 1
567.2.s.d 4 252.o even 6 1
567.2.s.d 4 252.bl odd 6 1
567.2.s.d 4 252.bn odd 6 1
1008.2.bt.b 4 7.c even 3 1
1008.2.bt.b 4 7.d odd 6 1
1008.2.bt.b 4 21.g even 6 1
1008.2.bt.b 4 21.h odd 6 1
1575.2.bc.a 8 140.w even 12 2
1575.2.bc.a 8 140.x odd 12 2
1575.2.bc.a 8 420.bp odd 12 2
1575.2.bc.a 8 420.br even 12 2
1575.2.bk.c 4 140.p odd 6 1
1575.2.bk.c 4 140.s even 6 1
1575.2.bk.c 4 420.ba even 6 1
1575.2.bk.c 4 420.be odd 6 1
7056.2.k.b 4 1.a even 1 1 trivial
7056.2.k.b 4 3.b odd 2 1 inner
7056.2.k.b 4 7.b odd 2 1 inner
7056.2.k.b 4 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(7056, [\chi])$$:

 $$T_{5}^{2} - 6$$ $$T_{11}^{2} + 2$$ $$T_{13}^{2} + 27$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -6 + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$( 2 + T^{2} )^{2}$$
$13$ $$( 27 + T^{2} )^{2}$$
$17$ $$( -24 + T^{2} )^{2}$$
$19$ $$( 3 + T^{2} )^{2}$$
$23$ $$( 32 + T^{2} )^{2}$$
$29$ $$( 8 + T^{2} )^{2}$$
$31$ $$( 3 + T^{2} )^{2}$$
$37$ $$( 1 + T )^{4}$$
$41$ $$( -54 + T^{2} )^{2}$$
$43$ $$( -1 + T )^{4}$$
$47$ $$( -150 + T^{2} )^{2}$$
$53$ $$( 8 + T^{2} )^{2}$$
$59$ $$( -24 + T^{2} )^{2}$$
$61$ $$( 12 + T^{2} )^{2}$$
$67$ $$( 11 + T )^{4}$$
$71$ $$( 50 + T^{2} )^{2}$$
$73$ $$( 3 + T^{2} )^{2}$$
$79$ $$( 5 + T )^{4}$$
$83$ $$( -54 + T^{2} )^{2}$$
$89$ $$( -24 + T^{2} )^{2}$$
$97$ $$( 108 + T^{2} )^{2}$$